Amenable groups with very poor compression into Lebesgue spaces

Abstract

We give a construction of finitely generated amenable groups that do not admit any coarse $1$-Lipschitz embedding with positive compression exponent into $L_p$ for any $1\le p<\infty$, including some that are four-step solvable, answering positively a question posed by Arzhantseva, Guba, and Sapir.